Growth optimal investment with transaction costs

For over thirty years mathematical economists have studied investing in financial markets. Most of the work on this subject has been limited to markets without costs. In this thesis we formulate growth optimal investment in markets with proportional transaction costs. We consider both continuous-time and discrete-time models of the market.;In the continuous time model we assume that the prices of the assets are log-normal. We show that the maximum achievable growth rate is a constant, independent of the initial endowment, and is given implicitly by a variational inequality. Using convexity we establish that the problem has an optimal policy. The optimal policy instantaneously adjusts the portfolio to the boundary of a no-trade set and then keeps the portfolio within this set by trading only when the portfolio hits the boundary. The variational inequality also shows that the no-trade set is a polyhedral cone in the gradient space. Next we give an asymptotic characterization of the no-trade set in the limit of small transaction costs. This asymptotic characterization leads to a simple investment strategy that compares well with the optimal one.;In the discrete time model the price relatives are assumed independent and identically distributed. The maximum achievable growth rate is again a constant, and is characterized by the associated Bellman equation. The problem has a control-limit optimal policy that moves the portfolio to the boundary of a no-trade set and thereafter takes action only when the portfolio moves out of the set. We consider two special discrete time markets: two-asset markets and horse race markets. For both these markets we construct a universal policy that learns the optimal policy "on the fly" and achieves the growth rate corresponding to the underlying unknown market distribution. Under some extra regularity conditions, we get a distribution free individual sequence result as well. For the horse race markets we derive an explicit expression for the maximum growth rate in terms of the entropy rate of the market.